New applications of random sampling in computational geometry
TL;DR: This paper gives several new demonstrations of the usefulness of random sampling techniques in computational geometry by creating a search structure for arrangements of hyperplanes by sampling the hyperplanes and using information from the resulting arrangement to divide and conquer.
Abstract: This paper gives several new demonstrations of the usefulness of random sampling techniques in computational geometry. One new algorithm creates a search structure for arrangements of hyperplanes by sampling the hyperplanes and using information from the resulting arrangement to divide and conquer. This algorithm requiresO(sd+?) expected preprocessing time to build a search structure for an arrangement ofs hyperplanes ind dimensions. The expectation, as with all expected times reported here, is with respect to the random behavior of the algorithm, and holds for any input. Given the data structure, and a query pointp, the cell of the arrangement containingp can be found inO(logs) worst-case time. (The bound holds for any fixed ?>0, with the constant factors dependent ond and ?.) Using point-plane duality, the algorithm may be used for answering halfspace range queries. Another algorithm finds random samples of simplices to determine the separation distance of two polytopes. The algorithm uses expectedO(n[d/2]) time, wheren is the total number of vertices of the two polytopes. This matches previous results [10] for the cased = 3 and extends them. Another algorithm samples points in the plane to determine their orderk Voronoi diagram, and requires expectedO(s1+?k) time fors points. (It is assumed that no four of the points are cocircular.) This sharpens the boundO(sk2 logs) for Lee's algorithm [21], andO(s2 logs+k(s?k) log2s) for Chazelle and Edelsbrunner's algorithm [4]. Finally, random sampling is used to show that any set ofs points inE3 hasO(sk2 log8s/(log logs)6) distinctj-sets withj≤k. (ForS ?Ed, a setS? ?S with |S?| =j is aj-set ofS if there is a half-spaceh+ withS? =S ?h+.) This sharpens with respect tok the previous boundO(sk5) [5]. The proof of the bound given here is an instance of a "probabilistic method" [15].
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01 Jul 1993
TL;DR: The tools used to obtain these results include Plu¨ckercoordinates of lines, random sampling and polarity transformations in3-space, and an expected time algorithm that tests the separation property.
Abstract: We study some combinatorial and algorithmic problems on sets of lines and polyhedral objects in 3-space. Our main results include: An O(n32c√log n) upper bound on the worst case complexity of the set of lines missing a star-shaped compact polyhedron with n edges.Given a tar-shaped compact polyhedron P with n edges we can compute on-line the shadow of P from a query direction v in almost-optimal ouput-sensitive time O(k log4 n), where k is the size of the shadow. The storage used by the data structure is O(n3+ϵ)An O(n32c√log n) upper bound on the worst case complexity of the set of lines that can be moved to infinity without the intersecting of a set of n given lines. This bound is almost tight.An O(n1.5 + ϵ) ranmdomized expected time algorithm that tests the separation property: there exists a direction v along which a set of n red lines can be tranlated away from a set of n blue lines without collsions?Computing the intersection of two polyhedral terrains in 3-space with n edges in time O(n4/3 + ϵ + k1/3n1+ϵ + k log2 n), where k is the size of the output, and ϵ > 0 an arbitrary small but fixed constant. This algorithm improves on the best previous result of the Chazelle at al. [7].The tools used to obtain these results include Plucker coordinates of lines, random sampling and polarity transformations in 3-space.
17 citations
Cites background from "New applications of random sampling..."
...From the random sampling theory [11], each simplex is cut by no more than O(n/r log r) of the Pliicker hyperplanes corresponding to lines in L....
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01 Jan 1996
TL;DR: Lower bounds on the number of primitive operations required to solve several fundamental problems in computational geometry are developed and it is proved that any partitioning algorithm requires $\Omega(\eta\sp{4/3}$) time to detect point-line incidences in the worst case.
Abstract: We develop lower bounds on the number of primitive operations required to solve several fundamental problems in computational geometry. For example, given a set of points in the plane, are any three colinear? Given a set of points and lines, does any point lie on a line? These and similar questions arise as subproblems or special cases of a large number of more complicated geometric problems, including point location, range searching, motion planning, collision detection, ray shooting, and hidden surface removal.
Previously these problems were studied only in general models of computation, but known techniques for these models are too weak to prove useful results. Our approach is to consider, for each problem, a more specialized model of computation that is still rich enough to describe all known algorithms for that problem. Thus, our results formally demonstrate inherent limitations of current algorithmic techniques. Our lower bounds dramatically improve previously known results and in most cases match known upper bounds, at least up to polylogarithmic factors.
In the first part of the thesis, we develop lower bounds for several degeneracy-detection problems, using adversary arguments. For example, we show that detecting colinear triples of points requires $\Omega(\eta\sp2$) sidedness queries in the worst case. Our lower bound follows from the construction of a set of points in general position with several "collapsible" triangles, any one of which can be made degenerate without changing the orientation of any other triangle. Using similar techniques, we prove lower bounds for deciding, given a set of points in $\IR\sp{\rm d}$, whether any d + 1 points lie on a hyperplane, whether any d + 2 points lie on a sphere, or whether the convex hull of the point is simplicial.
In the second part, we consider offline range searching problems, which are usually solved using geometric divide-and-conquer techniques. To study these problems, we introduce the class of partitioning algorithms. We prove that any partitioning algorithm requires $\Omega(\eta\sp{4/3}$) time to detect point-line incidences in the worst case. Using similar techniques, we prove an $\Omega(\eta\sp{4/3}$) lower bound for deciding if a set of points lies entirely above a set of hyperplanes in dimensions five and higher.
17 citations
Cites background or methods from "New applications of random sampling..."
...In higher dimensions, a randomized algorithm due to Clarkson [47] answers halfspace emptiness queries in time O(logn) after O(n) preprocessing time....
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...Partitioning algorithms for the halfspace emptiness problem can (and do [47, 107]) apply a version of the \containment shortcut" described in Section 6....
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01 Apr 1990
TL;DR: With this method, the recoverable coal is in physical contact with a series of two or more positive electrodes while it is isolated from a single negative electrode by an electrolytic solution.
Abstract: A electrolytic method and its associated apparatus for in situ recovery of coal products. With this method, the recoverable coal is in physical contact with a series of two or more positive electrodes while it is isolated from a single negative electrode by an electrolytic solution. The negative electrode and positive electrode are in boreholes which are drilled and are configured in the earth such that the positive electrodes may surround the single negative electrode. An electrolyte is placed in the center borehole with the negative electrode inserted in the liquid but insulated and positioned so as not to touch the coal in the side of the borehole. All of the positive electrodes in the adjacent boreholes are serially electrically connected to each other and to the positive terminal of an electrical potential difference source. The negative electrode is connected to the negative terminal of the same power source to initiate an electrolytic reaction in the coal-bearing earth. Coal products formed by this reaction may be periodically or continuously recovered and the extracted electrolyte replaced by a fresh electrolyte.
17 citations
Cites background from "New applications of random sampling..."
...The pre- cise bound is in terms of a certain 0 series, that can be associated with an arbitrary arrangement of hyper- planes....
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TL;DR: This work revisits the k-nearest-neighbor (k-NN) Voronoi diagram and presents a new paradigm for its construction, and introduces the K-NN Delaunay graph, which is the graph-theoretic dual of this diagram and is used as a base to directly compute this diagram in R2.
Abstract: We revisit the k-nearest-neighbor (k-NN) Voronoi diagram and present a new paradigm for its construction. We introduce the k-NN Delaunay graph, which is the graph-theoretic dual of the k-NN Voronoi diagram, and use it as a base to directly compute this diagram in R 2. We implemented our paradigm in the L 1 and L ? metrics, using segment-dragging queries, resulting in the first output-sensitive, O((n+m)logn)-time algorithm to compute the k-NN Voronoi diagram of n points in the plane, where m is the structural complexity (size) of this diagram. We also show that the structural complexity of the k-NN Voronoi diagram in the L ? (equiv. L 1) metric is O(min{k(n?k),(n?k)2}). Efficient implementation of our paradigm in the L 2 (resp. L p , 1
16 citations
08 Jul 2013
TL;DR: An algorithm to construct order-k Voronoi diagrams with a sweepline technique with O(nk2 log n) time complexity and O( nk) space complexity is presented.
Abstract: We present an algorithm to construct order-k Voronoi diagrams with a sweepline technique. The sites can be points or line segments. The algorithm has O(nk2 log n) time complexity and O(nk) space complexity.
16 citations
References
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Book•
01 Jan 1968
TL;DR: The arrangement of this invention provides a strong vibration free hold-down mechanism while avoiding a large pressure drop to the flow of coolant fluid.
Abstract: A fuel pin hold-down and spacing apparatus for use in nuclear reactors is disclosed. Fuel pins forming a hexagonal array are spaced apart from each other and held-down at their lower end, securely attached at two places along their length to one of a plurality of vertically disposed parallel plates arranged in horizontally spaced rows. These plates are in turn spaced apart from each other and held together by a combination of spacing and fastening means. The arrangement of this invention provides a strong vibration free hold-down mechanism while avoiding a large pressure drop to the flow of coolant fluid. This apparatus is particularly useful in connection with liquid cooled reactors such as liquid metal cooled fast breeder reactors.
17,939 citations
01 Jan 1985
TL;DR: This book offers a coherent treatment, at the graduate textbook level, of the field that has come to be known in the last decade or so as computational geometry.
Abstract: From the reviews: "This book offers a coherent treatment, at the graduate textbook level, of the field that has come to be known in the last decade or so as computational geometry...The book is well organized and lucidly written; a timely contribution by two founders of the field. It clearly demonstrates that computational geometry in the plane is now a fairly well-understood branch of computer science and mathematics. It also points the way to the solution of the more challenging problems in dimensions higher than two."
6,525 citations
"New applications of random sampling..." refers background in this paper
...(In fact the mapping γ is not unique in this regard: see [13, 23, 2]....
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Book•
01 Jan 1973
4,243 citations
TL;DR: This chapter reproduces the English translation by B. Seckler of the paper by Vapnik and Chervonenkis in which they gave proofs for the innovative results they had obtained in a draft form in July 1966 and announced in 1968 in their note in Soviet Mathematics Doklady.
Abstract: This chapter reproduces the English translation by B. Seckler of the paper by Vapnik and Chervonenkis in which they gave proofs for the innovative results they had obtained in a draft form in July 1966 and announced in 1968 in their note in Soviet Mathematics Doklady. The paper was first published in Russian as Вапник В. Н. and Червоненкис А. Я. О равномерноЙ сходимости частот появления событиЙ к их вероятностям. Теория вероятностеЙ и ее применения 16(2), 264–279 (1971).
3,939 citations
"New applications of random sampling..." refers background in this paper
...Vapnik and Chervonenkis [27] have derived general conditions under which several probabilities may be uniformly estimated using one random sample....
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Book•
01 Jan 1978TL;DR: In this article, the authors present a coherent treatment of computational geometry in the plane, at the graduate textbook level, and point out the way to the solution of the more challenging problems in dimensions higher than two.
Abstract: From the reviews: "This book offers a coherent treatment, at the graduate textbook level, of the field that has come to be known in the last decade or so as computational geometry...The book is well organized and lucidly written; a timely contribution by two founders of the field. It clearly demonstrates that computational geometry in the plane is now a fairly well-understood branch of computer science and mathematics. It also points the way to the solution of the more challenging problems in dimensions higher than two."
3,419 citations